Eigenvalues and simplicity of interval exchange transformations

Ferenczi, Sébastien; Zamboni, Luca Q.

Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 44 (2011) no. 3, pp. 361-392.

Pour une classe d’échanges de $d$ intervalles, que nous appelons la classe symétrique, nous définissons un nouveau processus d’induction autoduale, où le système est induit successivement sur des unions de sous-intervalles. Cet algorithme crée une structure de graphes qui reflète le comportement dynamique du système grâce aux tours de Rokhlin des applications induites. Nous l’utilisons pour construire un large choix d’exemples explicites sur quatre intervalles, avec différentes propriétés dynamiques  : on y trouve entre autres les premiers exemples non triviaux possédant des valeurs propres (rationnelles ou irrationnelles), le premier exemple d’un échange de plus de trois intervalles qui soit simple au sens de Veech (alors que cette notion, affaiblissant celle d’autocouplages minimaux, a été introduite en 1982 avec les échanges d’intervalles en vue), et un exemple inattendu qui est non uniquement ergodique, faiblement mélangeant pour une des mesures ergodiques invariantes, mais a des valeurs propres rationnelles pour l’autre mesure ergodique invariante.

For a class of $d$-interval exchange transformations, which we call the symmetric class, we define a new self-dual induction process in which the system is successively induced on a union of sub-intervals. This algorithm gives rise to an underlying graph structure which reflects the dynamical behavior of the system, through the Rokhlin towers of the induced maps. We apply it to build a wide assortment of explicit examples on four intervals having different dynamical properties: these include the first nontrivial examples with eigenvalues (rational or irrational), the first ever example of an exchange on more than three intervals satisfying Veech’s simplicity (though this weakening of the notion of minimal self-joinings was designed in 1982 to be satisfied by interval exchange transformations), and an unexpected example which is non uniquely ergodic, weakly mixing for one invariant ergodic measure but has rational eigenvalues for the other invariant ergodic measure.

Classification : 37A05, 37A25, 37B10
Keywords: interval exchanges, self-dual induction, eigenvalues, Veech simplicity
Mot clés : Échanges d'intervalles, induction autoduale, valeurs propres, simplicité au sens de Veech

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